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Details
Consider the lamellae-forming diblock copolymer melt which is swollen in a non-selective solvent. We confine ourselves by a symmetric copolymer consisting of N segments of the sort A and N segments of the sort B, N 1 ; each segment of the length a. If we assume that the fraction of the absorbed solvent is small enough (the case of the solvent vapor annealing), one can expect that the strong segregation regime is accessible for long enough blocks. The free energy of the nanostructured solution in this regime can be split into "long-range" elastic contribution of strongly stretched blocks (the end-to-end distance larger than that of the single chain in the solution), Fel, "short-range" entropic loses of the chains because of inhomogeneous monomer density at the interfaces, Finh, and the energy of interactions of monomer units and the solvent molecules, Fint. The elastic contribution of the blocks stretched perpendicular to the interfaces (along axis x) can be calculated within the Alexander-de Gennes model [] assuming equidistant location of the block's ends with respect to the interface. In the general case of inhomogeneous solvent distribution, the stretching of the blocks is also inhomogeneous. The local stretching E(x) can be calculated in a standard way [soft matter] through the dense packing condition for a thin lamellar layer of the thickness dx and lamellar area S, which is formed by Q chains, each of dn segments, dxS A ( x) B ( x) Qdnv :
E ( x) dx qv dn A ( x) B ( x)

(1)

Here volume

A, B

( x) are the volume fractions of A and B monomer units. Together with the solvent

fraction

S ( x) ,

they

are

subjected

to

the

incompressibility

condition,

S ( x) A ( x) B ( x) 1 . Excluded volume of the segment v of the flexible copolymer is
related to its length via v a3 ; q=Q/S is the number of chains (aggregation number) per unit area. The total elastic free energy per chain is obtained via integration of the local stretching with respect to x, H x H , where the AB interface is placed at x=0:
Fel 3 k B T 2a 3 dx Ndn dn 2a
N 2 H 2

2

H



dx

dx 3 dn 2a

H 2

H



dx

qv A ( x) B ( x)

(2)

Here, H is the end-to-end distance of the block (hemi-thickness of the A (B) layer). The last two contributions to the total free energy per chain can be approximated as:

Fin h Fin t 1 k BT qv

a 2 (A ) 2 a 2 (B ) 2 dx H 24 A 24 B F (A,B )
H

(3)

The entropic contribution (first two terms of the integral) is taken in conventional gradient form, (A, B )2 (A, B / x)2 (A, B )2 like for homopolymers. This approximation is accurate enough for diblocks in the strong segregation regime (it neglects connectivity of the blocks which gives small correction to the surface tension coefficient). The third term in the integral of eq 3 is the Flory-Huggins contribution describing interaction between components of the system as well as entropy of the solvent:


F (A , B ) 1 A B ln 1 A B ABAB 0 (A B )(1 A B )

(4)

Incompatibility between A and B monomer units are quantified by the parameter AB . The solvent is non-selective for the blocks. Therefore, the interactions between the solvent and different monomer units are described by one interaction parameter SA SB 0 . The average solvent concentration in the solution, S , is assumed to be fixed, S 1 Nqv / H . This assumption relates the end-to-end distance of the block and the aggregation number per unit area of the lamellae, H Nqv /(1 S ) . Finally, the total free energy of the solution takes the form

a2 dx 2 24 H H 1 H 1 H A dxA N B qv qv dxB N H H F 3 k BT 2a
H

dx

qv 1 A ( x) B ( x) qv

H

(A ) 2 a 2 (B ) 2 F (A , B ) A 24 B H N p qv 1 S

(5)

Last three terms in eq 5 introduce the space-filling conditions (normalization and relation between H and q; A , B and p are the Lagrange multipliers. Equil the polymer and solvent concentrations, as well as all other parameters of calculated via minimization of the functional with respect to the functions

A

of the densities) ibrium values of the system are ( x) , ( x) and
B

parameters H, q, A , B , and p (see Appendix for the details). In doing so, we exploited a
symmetry of the problem: A (0) B (0) , A ( x) B ( x) , A,B ( H ) 0 . The latter condition is a

consequence of a maximum (minimum) value of the concentration in the center of the corresponding layer and that the density has a smooth profile. For convenience of minimization, we introduce new variables,
2 2 A y A , B y B . Final set of equations, which has to be

solved numerically, has the following form:



H

H



a 2 2 a 2 2 3v 2 q dx y A yB 2 3 3 a
H H

2H

H



dx y y
2 A

2 B

1 q



1 dx y N q
2 A

H

H



dx y

2 B

3q v 2a 2 3q 2v 2a 2

22


2

y y

A

2 yA y B

22 A



a2 F A y y A y A 0 2 6 ( y A ) a2 F B y yB yB 0 2 6 ( yB )

(6)



2 yA y

22 A

H

qv N 1 S

where A B is due to the symmetry of the problem, A ( x) B ( x) .


Appendix
Minimization of the total free energy (5) with respect to the functions yA(x), yB(x), and parameters H, q, A , B , and p results in the following set of equations: yA:



3q 2v 2a 2

2


2

y

A

2 yA y

22 A





a2 F A y y A A y A 0 (A1) 2 6 ( y A )

yB:

3q 2v 2a 2



y

B

2 yA y

22 A

a2 F B y yB B yB 0 (A2) 2 6 ( yB )
(A3)

H:

3q 2 v 2 p 1 2 2 2 2 F ( y A ( H ), y B ( H )) A y A ( H ) B y B ( H ) 0 2 2 2 2 2a y A ( H ) y B ( H )
3q 2v 2a 2
2H

q:

H



dx 2 2 y A yB
H

H

H



dx a 2 2 a 2 2 2 2 2 2 yA yB F ( y A , yB ) A y A B yB pH 0 (A4) 6 6
1 q
H

A, B:

1 q

H



2 dx y A N ;

H



2 dx y B N

(A5)

p:

H

qv N 1 S
2 2 y A ( x) y B ( x )

(A6)

Using the symmetry of the problem,

, one can obtain

A ( x) B ( x) y y

via

taking the second derivative. Subtraction of (A1) from (A2) with the symmetry conditions gives

A B .
Multiplying (A1) and (A2) by y integrate the obtained equation:
A

and y B , respectively, and summarizing them, one can

3 q 2v 2 a2 a2 p 2 2 2 2 A ) 2 ( yB ) 2 F ( y A , yB ) A y A B yB E , (y 2 2 2a 2 y A y B 6 6 2
where the constant E is calculated with eq (A3) and eq (A7) at x=H:

(A7)

3 q 2v 2 2 2 2 2 F ( y A ( H ), yB ( H )) A y A ( H ) B yB ( H ) E , y 2 2 2 2a y A ( H ) y B ( H )

A, B

(H ) 0
(A8)

Finally, the first equation in the set of equations (6) is obtained via integration of eq (A7) with respect to x and combination with eq (A4).